In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace …
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems.
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family …
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results.
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison …
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications.
Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better …
Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.
This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions …
This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments …
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, R0 was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data.
In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving …
In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving electric circuits, population growth, vibrational beams, and heat conduction. The Mohanad transform is defined and introduced in this study, along with its fundamental qualities,including linearity and convolution. It is also discussed in connection with other integral transforms and how it is used in derivatives. Additionally, we use the Mohanad transform to solve a few systems of ordinary differential equations (ODEs) and review its properties in this paper. Determining the concentration of a chemical reactant (material) in a series is a physical chemistry problem that we use in the application part. We achieve this by developing a model based on ordinary differential equations (ODEs) and then solving them using the Mohanad transform. This research proves that, with little computational effort, we can get the exact solutions of ordinary differential equations (ODEs) via the Mohanad transform. We used graphs and tables to show our answer.
<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover …
<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover explicit solutions, including trigonometric, exponential, hyperbolic, and solitary wave solutions. Despite the extensive application of the Broer-Kaup model in tsunami wave analysis and plasma physics, existing literature has largely overlooked the complexity introduced by stochastic elements and fractional dimensions. Our study fills this critical gap by extending the traditional Broer-Kaup equations through the lens of stochastic forces, thereby offering a more comprehensive framework for analyzing hydrodynamic wave models. The novelty of our approach lies in the detailed investigation of the SBSE and SFBK equations, providing new insights into the behavior of shallow water waves under the influence of randomness. This work not only advances theoretical understanding but also enhances practical analysis capabilities by illustrating the effects of noise on wave propagation. Utilizing MATLAB for visual representation, we demonstrate the efficiency and flexibility of our method in addressing these sophisticated physical processes. The analytical solutions derived here mark a significant departure from previous findings, contributing novel perspectives to the field and paving the way for future research into complex wave dynamics.</p> </abstract>
<p>Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of …
<p>Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.</p>
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are …
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems.
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by …
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.
In this study, we investigate the $$(2+1)$$ -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal …
In this study, we investigate the $$(2+1)$$ -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal processing, geophysics, fluid dynamics, and plasma physics. Given its broad utility, a rigorous mathematical analysis of the JM equation is essential. The primary objective of this work is to derive exact soliton solutions using the Modified Sub-Equation (MSE) and Modified Auxiliary Equation (MAE) techniques. These solutions are computed using Maple 18, and encompass a variety of wave structures, including bright solitons, kink solitons, periodic waves, and singular solitons. The potential applications of these solutions span diverse domains, such as nonlinear dynamics, fiber optics, ocean engineering, software engineering, electrical engineering, and other areas of physical science. Through numerical simulations, we visualize the physical characteristics of the obtained soliton solutions using three distinct graphical formats: 3D surface plots, 2D contour plots, and line plots, based on the selection of specific parameter values. Our results demonstrate that the MSE and MAE techniques are not only efficient but also straightforward in extracting soliton solutions for the JM equation, outperforming other existing methods. Furthermore, the solutions presented in this study are novel, representing contributions that have not been previously reported in the literature.
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, …
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution.
This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen …
This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen is observed. The NABC method illustrates chaotic systems. We used the LNIM method to find analytical solutions for fractional Chen systems. The method stands out for its user-friendliness and numerical stability. The proposed methods are effective and yield analytical solutions and detection of chaotic behavior. Simultaneously, this results in a more precise understanding of the system. As a result, we may apply the approach to different systems and achieve more accurate findings. Furthermore, it has been demonstrated to be effective in accurately identifying instances through the exhibition of attractor chaos. Future applications in science and engineering can utilize these two methods to find numerical simulations and solutions to a variety of models.
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches …
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science.
In this manuscript, we analyze fuzzy-fixed-point results for fuzzy-mappings under some fuzzy contraction conditions in the setting of a complete fuzzy metric space. Fuzzy-fixed-point techniques are used in mathematical modeling …
In this manuscript, we analyze fuzzy-fixed-point results for fuzzy-mappings under some fuzzy contraction conditions in the setting of a complete fuzzy metric space. Fuzzy-fixed-point techniques are used in mathematical modeling to solve problems where traditional methods fail due to imprecise or uncertain data. To obtain fuzzy-fixed-points, different contraction conditions are implemented in a fuzzy context. To emphasize the impact of our research, we have furnished several intriguing examples. Applications are also incorporated to furnish the results. Previous results are given as corollaries from the relevant research. Our results extend and combine many results that exist in a significant area of related research.
In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and …
In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and cold plasma physics. We describe the fundamental properties of fractional calculus and its application to the STF-mBBM equation. Utilizing beta derivatives, we enhance our understanding of the intricate wave dynamics involved. Through the modified $$\left( \frac{G'}{G^{2}}\right)$$ -expansion method (M $$\left( \frac{G'}{G^{2}}\right)$$ -EM), we derive periodic, and kink singular soliton solutions and represent them graphically. We present the influence of the fractional parameter on traveling wave with 2D, 3D, surface and contour plots, providing a thorough understanding of the physical phenomena associated with the fractional model. In addition, we utilize the Hamiltonian property to analyze the chaotic dynamics of the solutions we've acquired. We perform two types of analysis using the Galilean transformation: a local sensitivity examination is conducted to see how the model responds to changes in individual input factors, and a global sensitivity examination is conducted to comprehend the correlation between the variability in the results and the variability in each input variable throughout its whole range of significance. This comprehensive approach allows us to determine traveling wave solutions effectively, offering new insights into the non-linear dynamical behavior of the system. The findings from this study are unique and significant for further exploration of the equation, offering valuable insights for future researchers.
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional …
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article.
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s …
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders α. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different α show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems.
This work presents a numerical method for solving fractional differential equations arising in mathematical biology. We specifically focus on two models: the fractional logistic growth model and the fractional SEIR …
This work presents a numerical method for solving fractional differential equations arising in mathematical biology. We specifically focus on two models: the fractional logistic growth model and the fractional SEIR model, which describe the population dynamics and the spread of epidemics, respectively. We use the fourth-order fractional Runge-Kutta (FRK4) method to approximate the solution of fractional differential equations (FDEs) associated with these models. The Caputo definition of the fractional derivative is used with our models, which is more appropriate for initial value problems involving real-life phenomena. The application of FRK4 provides a stable and accurate numerical solution for both models. The method is validated via numerical experiments, demonstrating efficiency and convergence in controlling fractional order dynamics. A comparative study with existing numerical techniques highlights the benefits of FRK4 in terms of accuracy and efficiency.
Solving nonlinear differential equations reliably and effectively is critical for applications in epidemiology, ecology, and finance. Traditional finite difference approaches frequently exhibit numerical instability and fail to maintain crucial features …
Solving nonlinear differential equations reliably and effectively is critical for applications in epidemiology, ecology, and finance. Traditional finite difference approaches frequently exhibit numerical instability and fail to maintain crucial features such as positivity and boundedness. Motivated by these restrictions, we provide an unconditionally stable Nonstandard Finite Difference Predictor-Corrector (NSFD-PC) strategy for solving nonlinear quadratic Riccati differential equations. Particularly when the step size h rises, NSFD-PC typically yields the solution that is closer to the exact solution. Riccati differential equation is 1st-order quadratic ode with several systems and control theory applications. Our technique achieves first-order temporal precision while keeping the continuous model's fundamental qualitative properties. We compare the proposed NSFD-PC scheme to the usual Euler Predictor-Corrector approach and find that it performs much better in accuracy and consistency with exact answers. The results show that our method is a superior and dependable option for solving nonlinear differential equations, making it extremely useful in various applications.
This paper presents a series of innovative Bipolar Neutrosophic Aggregation Operators to address the complexity and uncertainty in Multiple Criteria Decision-Making scenarios. The newly proposed operators BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, …
This paper presents a series of innovative Bipolar Neutrosophic Aggregation Operators to address the complexity and uncertainty in Multiple Criteria Decision-Making scenarios. The newly proposed operators BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG are designed to enhance aggregation under bipolar neutrosophic conditions, capturing a more nuanced view of decision-makers' preferences. We develop the MCDM method with the BNN. The effectiveness of these operators is demonstrated through a detailed case study, where they are applied to a complex decision-making scenario involving conflicting and uncertain criteria. Comparative and sensitivity analyses are conducted to assess the stability, reliability, and adaptability of each operator, benchmarking them against existing approaches. The results reveal that the proposed operators significantly improve decision-making accuracy by accommodating bipolar information and managing degrees of uncertainty more effectively. In the Results and Discussion section, we explore how each operator performs across varied MCDM contexts, highlighting the flexibility and robustness of the bipolar neutrosophic framework. The paper concludes by discussing the limitations of the proposed operators, offering insights into potential applications, and suggesting directions for future research to further refine bipolar neutrosophic-based MCDM approaches. This work contributes a comprehensive, operator-based method for enhanced decision-making under complex and uncertain conditions.
This work presents a numerical method for solving fractional differential equations arising in mathematical biology. We specifically focus on two models: the fractional logistic growth model and the fractional SEIR …
This work presents a numerical method for solving fractional differential equations arising in mathematical biology. We specifically focus on two models: the fractional logistic growth model and the fractional SEIR model, which describe the population dynamics and the spread of epidemics, respectively. We use the fourth-order fractional Runge-Kutta (FRK4) method to approximate the solution of fractional differential equations (FDEs) associated with these models. The Caputo definition of the fractional derivative is used with our models, which is more appropriate for initial value problems involving real-life phenomena. The application of FRK4 provides a stable and accurate numerical solution for both models. The method is validated via numerical experiments, demonstrating efficiency and convergence in controlling fractional order dynamics. A comparative study with existing numerical techniques highlights the benefits of FRK4 in terms of accuracy and efficiency.
Solving nonlinear differential equations reliably and effectively is critical for applications in epidemiology, ecology, and finance. Traditional finite difference approaches frequently exhibit numerical instability and fail to maintain crucial features …
Solving nonlinear differential equations reliably and effectively is critical for applications in epidemiology, ecology, and finance. Traditional finite difference approaches frequently exhibit numerical instability and fail to maintain crucial features such as positivity and boundedness. Motivated by these restrictions, we provide an unconditionally stable Nonstandard Finite Difference Predictor-Corrector (NSFD-PC) strategy for solving nonlinear quadratic Riccati differential equations. Particularly when the step size h rises, NSFD-PC typically yields the solution that is closer to the exact solution. Riccati differential equation is 1st-order quadratic ode with several systems and control theory applications. Our technique achieves first-order temporal precision while keeping the continuous model's fundamental qualitative properties. We compare the proposed NSFD-PC scheme to the usual Euler Predictor-Corrector approach and find that it performs much better in accuracy and consistency with exact answers. The results show that our method is a superior and dependable option for solving nonlinear differential equations, making it extremely useful in various applications.
This paper presents a series of innovative Bipolar Neutrosophic Aggregation Operators to address the complexity and uncertainty in Multiple Criteria Decision-Making scenarios. The newly proposed operators BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, …
This paper presents a series of innovative Bipolar Neutrosophic Aggregation Operators to address the complexity and uncertainty in Multiple Criteria Decision-Making scenarios. The newly proposed operators BNAAWA, BNAAOWA, BNAAHWA, BNAAWG, BNAAOWG, and BNAAHWG are designed to enhance aggregation under bipolar neutrosophic conditions, capturing a more nuanced view of decision-makers' preferences. We develop the MCDM method with the BNN. The effectiveness of these operators is demonstrated through a detailed case study, where they are applied to a complex decision-making scenario involving conflicting and uncertain criteria. Comparative and sensitivity analyses are conducted to assess the stability, reliability, and adaptability of each operator, benchmarking them against existing approaches. The results reveal that the proposed operators significantly improve decision-making accuracy by accommodating bipolar information and managing degrees of uncertainty more effectively. In the Results and Discussion section, we explore how each operator performs across varied MCDM contexts, highlighting the flexibility and robustness of the bipolar neutrosophic framework. The paper concludes by discussing the limitations of the proposed operators, offering insights into potential applications, and suggesting directions for future research to further refine bipolar neutrosophic-based MCDM approaches. This work contributes a comprehensive, operator-based method for enhanced decision-making under complex and uncertain conditions.
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s …
This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders α. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different α show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems.
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional …
On the one hand, convex functions are important to derive rigorous convergence rates, and on the other, synchronous functions are significant to solve statistical problems using Chebyshev inequalities. Therefore, fractional integral inequalities involving such functions play a crucial role in creating new models and methods. Although a large class of fractional operators have been used to establish inequalities, nevertheless, these operators having the Fox-H and the Meijer-G functions in their kernel have been applied to establish fractional integral inequalities for such important classes of functions. Taking motivation from these facts, the primary objective of this work is to develop fractional inequalities involving the Fox-H function for convex and synchronous functions. Since the Fox-H function generalizes several important special functions of fractional calculus, our results are significant to innovate the existing literature. The inventive features of these functions compel researchers to formulate deeper results involving them. Therefore, compared with the ongoing research in this field, our results are general enough to yield novel and inventive fractional inequalities. For instance, new inequalities involving the Meijer-G function are obtained as the special cases of these outcomes, and certain generalizations of Chebyshev inequality are also included in this article.
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches …
In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science.
In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and …
In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and cold plasma physics. We describe the fundamental properties of fractional calculus and its application to the STF-mBBM equation. Utilizing beta derivatives, we enhance our understanding of the intricate wave dynamics involved. Through the modified $$\left( \frac{G'}{G^{2}}\right)$$ -expansion method (M $$\left( \frac{G'}{G^{2}}\right)$$ -EM), we derive periodic, and kink singular soliton solutions and represent them graphically. We present the influence of the fractional parameter on traveling wave with 2D, 3D, surface and contour plots, providing a thorough understanding of the physical phenomena associated with the fractional model. In addition, we utilize the Hamiltonian property to analyze the chaotic dynamics of the solutions we've acquired. We perform two types of analysis using the Galilean transformation: a local sensitivity examination is conducted to see how the model responds to changes in individual input factors, and a global sensitivity examination is conducted to comprehend the correlation between the variability in the results and the variability in each input variable throughout its whole range of significance. This comprehensive approach allows us to determine traveling wave solutions effectively, offering new insights into the non-linear dynamical behavior of the system. The findings from this study are unique and significant for further exploration of the equation, offering valuable insights for future researchers.
In this manuscript, we analyze fuzzy-fixed-point results for fuzzy-mappings under some fuzzy contraction conditions in the setting of a complete fuzzy metric space. Fuzzy-fixed-point techniques are used in mathematical modeling …
In this manuscript, we analyze fuzzy-fixed-point results for fuzzy-mappings under some fuzzy contraction conditions in the setting of a complete fuzzy metric space. Fuzzy-fixed-point techniques are used in mathematical modeling to solve problems where traditional methods fail due to imprecise or uncertain data. To obtain fuzzy-fixed-points, different contraction conditions are implemented in a fuzzy context. To emphasize the impact of our research, we have furnished several intriguing examples. Applications are also incorporated to furnish the results. Previous results are given as corollaries from the relevant research. Our results extend and combine many results that exist in a significant area of related research.
In this study, we investigate the $$(2+1)$$ -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal …
In this study, we investigate the $$(2+1)$$ -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal processing, geophysics, fluid dynamics, and plasma physics. Given its broad utility, a rigorous mathematical analysis of the JM equation is essential. The primary objective of this work is to derive exact soliton solutions using the Modified Sub-Equation (MSE) and Modified Auxiliary Equation (MAE) techniques. These solutions are computed using Maple 18, and encompass a variety of wave structures, including bright solitons, kink solitons, periodic waves, and singular solitons. The potential applications of these solutions span diverse domains, such as nonlinear dynamics, fiber optics, ocean engineering, software engineering, electrical engineering, and other areas of physical science. Through numerical simulations, we visualize the physical characteristics of the obtained soliton solutions using three distinct graphical formats: 3D surface plots, 2D contour plots, and line plots, based on the selection of specific parameter values. Our results demonstrate that the MSE and MAE techniques are not only efficient but also straightforward in extracting soliton solutions for the JM equation, outperforming other existing methods. Furthermore, the solutions presented in this study are novel, representing contributions that have not been previously reported in the literature.
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments …
In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects, R0 was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data.
This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen …
This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen is observed. The NABC method illustrates chaotic systems. We used the LNIM method to find analytical solutions for fractional Chen systems. The method stands out for its user-friendliness and numerical stability. The proposed methods are effective and yield analytical solutions and detection of chaotic behavior. Simultaneously, this results in a more precise understanding of the system. As a result, we may apply the approach to different systems and achieve more accurate findings. Furthermore, it has been demonstrated to be effective in accurately identifying instances through the exhibition of attractor chaos. Future applications in science and engineering can utilize these two methods to find numerical simulations and solutions to a variety of models.
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by …
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.
In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving …
In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving electric circuits, population growth, vibrational beams, and heat conduction. The Mohanad transform is defined and introduced in this study, along with its fundamental qualities,including linearity and convolution. It is also discussed in connection with other integral transforms and how it is used in derivatives. Additionally, we use the Mohanad transform to solve a few systems of ordinary differential equations (ODEs) and review its properties in this paper. Determining the concentration of a chemical reactant (material) in a series is a physical chemistry problem that we use in the application part. We achieve this by developing a model based on ordinary differential equations (ODEs) and then solving them using the Mohanad transform. This research proves that, with little computational effort, we can get the exact solutions of ordinary differential equations (ODEs) via the Mohanad transform. We used graphs and tables to show our answer.
<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover …
<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover explicit solutions, including trigonometric, exponential, hyperbolic, and solitary wave solutions. Despite the extensive application of the Broer-Kaup model in tsunami wave analysis and plasma physics, existing literature has largely overlooked the complexity introduced by stochastic elements and fractional dimensions. Our study fills this critical gap by extending the traditional Broer-Kaup equations through the lens of stochastic forces, thereby offering a more comprehensive framework for analyzing hydrodynamic wave models. The novelty of our approach lies in the detailed investigation of the SBSE and SFBK equations, providing new insights into the behavior of shallow water waves under the influence of randomness. This work not only advances theoretical understanding but also enhances practical analysis capabilities by illustrating the effects of noise on wave propagation. Utilizing MATLAB for visual representation, we demonstrate the efficiency and flexibility of our method in addressing these sophisticated physical processes. The analytical solutions derived here mark a significant departure from previous findings, contributing novel perspectives to the field and paving the way for future research into complex wave dynamics.</p> </abstract>
<p>Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of …
<p>Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.</p>
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family …
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results.
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are …
In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems.
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, …
A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution.
Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better …
Visceral leishmaniosis is one recent example of a global illness that demands our best efforts at understanding. Thus, mathematical modeling may be utilized to learn more about and make better epidemic forecasts. By taking into account the Caputo and Caputo-Fabrizio derivatives, a frictional model of visceral leishmaniosis was mathematically examined based on real data from Gedaref State, Sudan. The stability analysis for Caputo and Caputo-Fabrizio derivatives is analyzed. The suggested ordinary and fractional differential mathematical models are then simulated numerically. Using the Adams-Bashforth method, numerical simulations are conducted. The results demonstrate that the Caputo-Fabrizio derivative yields more precise solutions for fractional differential equations.
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison …
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications.
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace …
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems.
This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions …
This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace …
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems.
This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have …
This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.
In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional …
In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.
The subject of this study is the solution of a fractional Bernoulli equation and a chaotic system by using a novel scheme for the fractional derivative and comparison of approximate …
The subject of this study is the solution of a fractional Bernoulli equation and a chaotic system by using a novel scheme for the fractional derivative and comparison of approximate and exact solutions. It is found that the suggested method produces solutions that are identical to the exact solution. We can therefore generalize the strategy to different systems to get more accurate results. We think that the novel fractional derivative scheme that has been offered and the algorithm that has been suggested will be utilized in the future to construct and simulate a variety of fractional models that can be used to solve more difficult physics and engineering challenges.
The primary goal of this study is to provide a novel mathematical model for Influenza using the Atangana–Baleanu Caputo fractional-order derivative operator (ABC-Operator) in place of the standard operator. There …
The primary goal of this study is to provide a novel mathematical model for Influenza using the Atangana–Baleanu Caputo fractional-order derivative operator (ABC-Operator) in place of the standard operator. There will be an examination of how the influenza-positive solutions reacts to real-world data. The fractional Euler Method will be utilized to reveal the dynamics of the influenza mathematical model. Both the stability of the disease-free equilibrium and the endemic equilibrium points, two symmetrical extrema of the proposed dynamical model, are examined. It will be shown, using numerical comparisons, that the findings obtained by employing the fractional-order model are considerably more similar to certain actual data than the integer-order model's results. These should shed light on the significance of fractional calculus when confronting epidemic risks.
This study provides a comprehensive exploration of the qualitative analysis of a hybrid system of pantograph equations with fractional order and a p-Laplacian operator. The existence of the solution of …
This study provides a comprehensive exploration of the qualitative analysis of a hybrid system of pantograph equations with fractional order and a p-Laplacian operator. The existence of the solution of the system is explicitly established within the context of Riemann–Liouville's fractional order operator, employing the Arzelà–Ascoli theorem for validation. The establishment of uniqueness criteria is accomplished by the utilization of the Banach contractive technique. In addition, the examination of solution stability is conducted using the Hyers–Ulam (HU) stability technique. In order to enhance the credibility of our main conclusions, we have included a representative and illustrative example in the concluding section of the study. This work serve to offer a thorough and applicable comprehension of the mathematical framework that has been proposed.
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. …
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton approach. Numerical simulations and figures are presented and discussed, in order to show the efficiency of the proposed method. In the future, we anticipate that the provided generalized Caputo fractional derivative and the suggested method will be utilized to create and simulate a wide variety of generalized Caputo-type fractional models. We have included examples to demonstrate the accuracy of the present method.
In this article, we use the <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mi mathvariant="script">p</mi></math> -Laplace decomposition method to find the solution to the initial value problems that involve generalized fractional derivatives. The <math xmlns="http://www.w3.org/1998/Math/MathML" …
In this article, we use the <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"><mi mathvariant="script">p</mi></math> -Laplace decomposition method to find the solution to the initial value problems that involve generalized fractional derivatives. The <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"><mi mathvariant="script">p</mi></math> -Laplace decomposition method is used to get approximate series solutions. The Adomian decomposition is improved with the assistance of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"><mi mathvariant="script">p</mi></math> -Laplace transform to examine the solutions of the given examples to demonstrate the precision of the current technique.
The emergence of multi-disease epidemics presents an escalating threat to global health. In response to this serious challenge, we present an innovative stochastic susceptible–vaccinated–infected–recovered epidemic model that addresses the dynamics …
The emergence of multi-disease epidemics presents an escalating threat to global health. In response to this serious challenge, we present an innovative stochastic susceptible–vaccinated–infected–recovered epidemic model that addresses the dynamics of two diseases alongside intricate vaccination strategies. Our novel model undergoes a comprehensive exploration through both theoretical and numerical analyses. The stopping time concept, along with appropriate Lyapunov functions, allows us to explore the possibility of a globally positive solution. Through the derivation of reproduction numbers associated with the stochastic model, we establish criteria for the potential extinction of the diseases. The conditions under which one or both diseases may persist are explained. In the numerical aspect, we derive a computational scheme based on the Milstein method. The scheme will not only substantiate the theoretical results but also facilitate the examination of the impact of parameters on disease dynamics. Through examples and simulations, we have a crucial impact of varying parameters on the system’s behavior.
In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation …
In this article, we study and investigate the analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov-Kuznetsov (mKDV-ZK) equation. We have got new exact solutions of the fractional mKDV-ZK equation by using first integral method; we found new types of hyperbolic solutions and trigonometric solutions by symbolic computation.
Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena. Phase transitions between the fractional- and integer-order cases is …
Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena. Phase transitions between the fractional- and integer-order cases is one of the main problems that have been extensively examined by scientists, economists, and engineers. This paper reports the existence of chaotic attractors that exist only in the fractional-order case when using the specific selection of parameter values in a new hyperchaotic (Matouk's) system. This paper discusses stability analysis of the steady-state solutions, existence of hidden chaotic attractors and self-excited chaotic attractors. The results are supported by computing basin sets of attractions, bifurcation diagrams and the Lyapunov exponent spectrum. These tools verify the existence of chaotic dynamics in the fractional-order case; however, the corresponding integer-order counterpart exhibits quasi-periodic dynamics when using the same choice of initial conditions and parameter set. Projective synchronization via non-linear controllers is also achieved between drive and response states of the hidden chaotic attractors of the fractional Matouk's system. Dynamical analysis and computer simulation results verify that the chaotic attractors exist only in the fractional-order case when using the specific selection of parameter values in the Matouk's hyperchaotic system. An example of the existence of hidden and self-excited chaotic attractors that appears only in the fractional-order case is discussed. So, the obtained results give the first example that shows chaotic states are not necessarily transmitted between fractional- and integer-order dynamical systems when using a specific selection of parameter values. Chaos synchronization using the hidden attractors' manifolds provides new challenges in chaos-based applications to technology and industrial fields.
This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new …
This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.
Hirota’s bilinear method (HBM) has been successfully applied to the [Formula: see text]-dimensional Pavlov equation to analyze the different wave structures in this paper. The [Formula: see text]-dimensional Pavlov equation …
Hirota’s bilinear method (HBM) has been successfully applied to the [Formula: see text]-dimensional Pavlov equation to analyze the different wave structures in this paper. The [Formula: see text]-dimensional Pavlov equation is used for the study of integrated hydrodynamic chains and Einstein–Weyl manifolds. In our research, we find new solutions in the forms of lump solutions, breather waves, and two-wave solutions. The modulation instability (MI) of the governing model is also discussed. Moreover, a variety of 3D, 2D, and contour profiles are used to illustrate the physical behavior of the reported results. Acquired findings are useful in understanding nonlinear science and its related nonlinear higher-dimensional wave fields. Through the use of Mathematica, the obtained results are verified by inserting them into the governing equation. The strengthening of representative calculations we’ve made gives us a strong and effective mathematical framework for dealing with the most difficult nonlinear wave problems.
The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse …
The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.
In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and …
In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and the uniqueness of result is established by applying Banach contraction mapping principle. An example is offered to ensure the validity of our obtained results.
In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. …
In this article, the iteration transform method is used to evaluate the solution of a fractional-order dark optical soliton, bright optical soliton, and periodic solution of the nonlinear Schrödinger equations. The Caputo operator describes the fractional-order derivatives. The solutions of some illustrative examples are presented to show the validity of the proposed technique without using any polynomials. The proposed method provides the series form solutions, which converge to the exact results. Using the present methodology, the solutions of fractional-order problems as well as integral-order problems are calculated. The present method has less computational costs and a higher rate of convergence. Therefore, the suggested algorithm is constructive to solve other fractional-order linear and nonlinear partial differential equations.
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison …
This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method (ρ-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications.
The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When …
The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
In this paper, we have introduced the analytical solutions of the Benjamin-Bona-Mahony equation and the (2+1) dimensional breaking soliton equations with the help of a new Algorithm of first integral …
In this paper, we have introduced the analytical solutions of the Benjamin-Bona-Mahony equation and the (2+1) dimensional breaking soliton equations with the help of a new Algorithm of first integral method formula two (AFIM), by depending on mathematical software’s. New and more general variety of families of exact solutions have been represented by different structures of 3rd dimension plotting and contouring plotting with different parameters. So, the solution in this research is unique, new and more general. We can apply in computer sciences, mathematical physics, with a different vision, general and Programmable.
Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In …
Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In this study, we highlight the stochastic fractional space Allen-Cahn equation (SFACE) as a major application of this class. In addition, we utilize the simplest equation method (SEM) with a dual sense of Brownian motion to convert the presented equation into an ordinary differential equation (ODE) and apply an effective computational technique to obtain exact solutions. By carefully comparing the derived solutions with solutions from other articles, we prove the distinction of these solutions for their diversity and the discovery of new solutions for SFACE that appear in many scientific fields, such as mathematical biology, quantum mechanics, and plasma physics. The results introduced in this article were obtained by plotting several graphs and examining how noise affects exact solutions using Mathematica and MATLAB software packages.
The analytical solutions for an important generalized Nonlinear evolution equations NLEEs dynamical partial differential equations (DPDEs) that involve independent variables represented by the (2 + 1)-dimensional breaking soliton equation, the …
The analytical solutions for an important generalized Nonlinear evolution equations NLEEs dynamical partial differential equations (DPDEs) that involve independent variables represented by the (2 + 1)-dimensional breaking soliton equation, the (2 + 1)-dimensional Calogero--Bogoyavlenskii--Schiff (CBS) equation, and the (2 +1)-dimensional Bogoyavlenskii's breaking soliton equation (BE), and some new exact propagating solutions to a generalized (3+1)-dimensional KP equation with variable coefficients are constructed by using a new algorithm of the first integral method (NAFIM) and determined some analytical solutions by appointing special values of the parameters. In addition to that, we showed a new variety and unique travelling wave solutions by graphical illustration with symbolic computations.
In this paper, we solve a nonlinear fractional-order model for analyzing the dynamical behavior of vector-borne diseases within the frame of Caputo-fractional derivative. The proposed mathematical model advances the existing …
In this paper, we solve a nonlinear fractional-order model for analyzing the dynamical behavior of vector-borne diseases within the frame of Caputo-fractional derivative. The proposed mathematical model advances the existing integer-order model on transmission and cure of vector-borne diseases. The existence and uniqueness of the solutions of the fractional-order model are proved using the Banach contraction principle. We investigate the local asymptomatic stability for the obtained disease-free equilibrium point and global stability for the proposed model in the sense of Ulam–Hyers stability criteria, respectively. Besides that, we obtain a numerical solution for the projected model using the Corrector-Predictor algorithm. Finally, to illustrate the obtained theoretical results, we perform numerical simulations for different values of fractional-order derivative and make a comparison with the results of the integer-order derivative.
These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important …
These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important procedures in preventing the spread of infectious diseases, as incorrect testing can turn a susceptible person into an infected person. In this paper, we study the dynamics of imperfect testing and diagnostics of infectious diseases model by replacing the integer order derivative in the sense of fractional order Atangana–Baleanu operator coupled with Caputo operator. This AB-fractional operator is the generalization of classical derivative and gives more data of the factors of the nonlocal dynamical frameworks. Fixed point theorems have been used for the verification of existence results and Picard’s stability technique utilized for the stability study of a fractional order imperfect testing infectious disease (ITID) model. Finally, numerical computations are implemented for the fractional order ITID model to illustrate the results graphically.
Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able …
Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.
In this paper, we implement a Functional Variable Method (FVM) for extracting various kind of exact traveling wave solutions of the time-space fractional-order generalized Zakharov equation (FGZE). The method is …
In this paper, we implement a Functional Variable Method (FVM) for extracting various kind of exact traveling wave solutions of the time-space fractional-order generalized Zakharov equation (FGZE). The method is extremely simple and effective for handling nonlinear equations with fractional derivatives arising in mathematical physics.
In this paper, the Benettin–Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. First, it …
In this paper, the Benettin–Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. First, it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams–Bashforth–Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams–Bashforth–Moulton method, is utilized. Four representative examples are considered.
Recently, fractional calculus theory has been successfully employed in generalized thermoelasticity theory and several models with fractional order have been introduced. In this work, a fractional thermoelastic modified Fourier's Law …
Recently, fractional calculus theory has been successfully employed in generalized thermoelasticity theory and several models with fractional order have been introduced. In this work, a fractional thermoelastic modified Fourier's Law with phase lag and two different fractional-orders has been constructed. The previous fractional models of thermoelasticity introduced by Sherief et al. [1], Ezzat [2] and Lord and Shulman [3] as well as classical coupled thermoelasticity [4] follow as limiting cases. This proposed model is applied to an infinitely annular cylinder that is subject to time-dependent surface temperatures, and whose surfaces are free of traction. The method of the Laplace transform is employed to get the solutions of the field variables. A numerical technique is utilized to invert the Laplace transforms. Some results are presented in tables and figures to extract the effects of fractional order parameters on all variables studied. The theory's predictions have been checked and compared to previous models.
In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently …
In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense. This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions. Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models. The validity of the present algorithm is illustrated by expounding and testing two numerical examples. The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering.
We apply a new generalized Caputo operator to investigate the dynamical behaviour of the non-integer food web model (FWM). This dynamical model has three population species and is nonlinear. Three …
We apply a new generalized Caputo operator to investigate the dynamical behaviour of the non-integer food web model (FWM). This dynamical model has three population species and is nonlinear. Three types of species are considered in this population: prey species, intermediate predators, and top predators, and the top predators are also divided into mature and immature predators. We calculated the uniqueness and existence of the solutions applying the fixed-point hypothesis. Our study examines the possibility of obtaining new dynamical phase portraits with the new generalized Caputo operator and demonstrates the portraits for several values of fractional order. A generalized predictor–corrector (P-C) approach is utilized in numerically solving this food web model. In the case of the nonlinear equations system, the effectiveness of the used scheme is highly evident and easy to implement. In addition, stability analysis was conducted for this numerical scheme.
The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Jimbo-Miwa-like (JM) equation, the extended (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, …
The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Jimbo-Miwa-like (JM) equation, the extended (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, the generalization of the (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>)-dimensional Bogoyavlensky-Konopelchenko (BK) equation, and a variable-coefficient extension of the DJKM (vDJKM) equation, which contain one-soliton-, two-soliton-, and triple-soliton-kind solutions. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.
Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for …
Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the …
Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.
The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, \[ \big({}^ρ\mathcal{I}^α_{a+}f\big)(x) = \frac{ρ^{1- α}}{Γ(α)} \int^x_a \frac{τ^{ρ-1} f(τ) }{(x^ρ- τ^ρ)^{1-α}}\, dτ, \] which generalizes the …
The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, \[ \big({}^ρ\mathcal{I}^α_{a+}f\big)(x) = \frac{ρ^{1- α}}{Γ(α)} \int^x_a \frac{τ^{ρ-1} f(τ) }{(x^ρ- τ^ρ)^{1-α}}\, dτ, \] which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.